Question - Solving Absolute Value Inequalities
Solution:
The inequality $$ |t - 75| \leq 15 $$ represents the values of $$ t $$ that are at most 15 units away from 75 on the number line, either less than or equal to 75 or greater than or equal to 75, but not further away than 15 units in either direction. To solve this absolute value inequality, we'll break it into two separate inequalities:1. $$ t - 75 \leq 15 $$2. $$ - (t - 75) \leq 15 $$ which is the same as $$ t - 75 \geq -15 $$Now, we solve each inequality:1. $$ t \leq 15 + 75 $$ $$ t \leq 90 $$2. $$ t - 75 \geq -15 $$ $$ t \geq -15 + 75 $$ $$ t \geq 60 $$Combining these two inequalities, we get:\[ 60 \leq t \leq 90 \]Therefore, the solution to the absolute value inequality is all real numbers $$ t $$ between 60 and 90, inclusive.
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